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ODE's

Many dynamical systems in physics, astronomy, chemistry, physiology, meteorology, economics, population
dynamics can be described by
Ordinary Differential Equations.
In the second half of the 20th century much attention has been focussed on the
often
chaotic,
i.e. unpredictable behaviour of (non-linear) ODE's. A well-known example is the
Lorenz atrractor, illustrating the "Butterfly effect": small causes can have large effects.
Mathgrapher uses an accurate Adams-Bashforth variable order, variable step predictor-corrector
algorithm to integrate systems of up to 20 coupled differential equations.

Lorenz attractor (the "Butterfly effect")
Lorenz studied a set of differential equations that are well known in
hydrodynamics (the Navier-Stokes quations in combination with heat
conduction equation and the continuity equation) which are used for example
in numerical weather simulation programs. He came up with a simple (though non-linear)
set of ODE's which turned out to exhibit very remarkable behaviour. One
of the demonstrations in Mathgrapher shows the integration of this system
and some analysis of the results. The 3D orbit looks like a butterfly.
The wings of the butterfly may represent two quite different states of
the system like a thunderstorm and calm sunny weather. When the
evolution of the system (the orbit) is studied it turns out that the system switches
from one wing of the butterfly to the other in a chaotic way. The behaviour
turns out to be highly sensitive to the initial conditions and therefore
becomes highly unpredicatable or chaotic. Lorenz remarked that "the flapping
of a butterfly's wings could change the weather".
The orbit in the Lorenz system of ODE's remains confined to a definite
volume in space. It appears as though the trajectory is attracted to a
certain region in space. This is called the "Lorenz attractor". The Lorenz
attractor even belongs to the category of "strange attractors" which exhibit chaotic
behaviour.

Y-Z projection of the Lorenz attractor

The chaotic behaviour becomes clear when one looks at the power spectrum analysis
of the orbit.